# Use or Importance of $\Rightarrow^*$ operator

Particularly, there are 2 variants of $$\Rightarrow$$, one is $$\Rightarrow^*$$ and another is $$\Rightarrow^+$$ which are used in derivation of strings using the productions of the grammar.

As stated here, when $$\Rightarrow^*$$ is used, it means derive in 0 or more steps and when $$\Rightarrow^+$$ is used, it means derive in 1 or more steps.

What I don't understand is what is the use or importance of $$\Rightarrow^*$$ operator?. It just replaces the head of the production with the head itself.
Is it that the $$\Rightarrow^*$$ operator is just there to define operators in some standard way or it actually differs in the way it is used as compared to $$\Rightarrow^+$$?

Also, while deriving strings, to skip some obivious steps(like replacing multiple non-terminals in a string with their respective terminals), should I use $$\Rightarrow^*$$ or $$\Rightarrow^+$$?

It is false that $$\Rightarrow^*$$ " just replaces the head of the production with the head itself". $$A \Rightarrow^* B$$ means that the sentential form $$B$$ can be obtained from the sentimental form $$A$$ by applying any number of productions (which might be $$0$$ or more than $$0$$).
For example if your grammar is $$S \to a$$, then all of the following are true $$S \Rightarrow^* S$$, $$S \Rightarrow^* a$$, and $$S \Rightarrow^+ a$$, while it is false that $$S \Rightarrow^+ S$$.
If you are deriving strings and applying at least one production per step then both $$\Rightarrow^*$$ and $$\Rightarrow^+$$ are correct. The latter is more specific since it implies the former (but not vice-versa).
• According to you, "the sentential form B can be obtained from the sentential form A by applying any number of productions (which might be 0 or more than 0)", so in case of applying no productions while deriving a new string, I infer it means that the new derived string will be same as before. So, if my inference is correct then, what is the significance of this reflexive property of $\Rightarrow^*$ operator? – Perspicacious Apr 27 '20 at 9:45
• $A \Rightarrow^* B$ is just a notation to mean that $B$ can be derived from $A$ in some way. You use it when you want to express that... Consider a sample grammar $G$ defined as follows $S \to A|B, \quad A \to Aa | a, \quad B \to AA$. The following statement would be true: $\forall$ sentential form $X$, there exist a sentence $y \in a^*$ such that $X \Rightarrow^* y$. But the following statement would be false: $\forall$ sentential form $X$, there exist a sentence $y \in a^*$ such that $X \Rightarrow^+ y$. – Steven Apr 27 '20 at 11:29
• ok, so, does it mean the operator $\Rightarrow^*$ is required (if that's the correct word) to have a more comprehensive definition of the symbols?? – Perspicacious Apr 27 '20 at 11:57